If `A=[{:(3,1),(2,-3):}]`, then find `|adj.A|`.

Answer

To find \(|\text{adj} A|\) for the matrix \(A = \begin{pmatrix} 3 & 1 \\ 2 & -3 \end{pmatrix}\), we will follow these steps:

### Step 1: Find the Determinant of Matrix \(A\)
The determinant of a \(2 \times 2\) matrix \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is calculated using the formula:
\[ |A| = ad - bc \]
For our matrix \(A\):
- \(a = 3\)
- \(b = 1\)
- \(c = 2\)
- \(d = -3\)

Now, substituting these values into the determinant formula:
\[ |A| = (3)(-3) - (1)(2) = -9 - 2 = -11 \]

### Step 2: Find the Adjoint of Matrix \(A\)
The adjoint of a \(2 \times 2\) matrix is given by:
\[ \text{adj} A = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
Substituting the values from matrix \(A\):
\[ \text{adj} A = \begin{pmatrix} -3 & -1 \\ -2 & 3 \end{pmatrix} \]

### Step 3: Find the Determinant of the Adjoint of Matrix \(A\)
The determinant of the adjoint of a \(2 \times 2\) matrix can be calculated using the same formula:
\[ |\text{adj} A| = |d \cdot a - (-b)(-c)| \]
Substituting the values from the adjoint matrix:
- \(d = -3\)
- \(a = 3\)
- \(b = -1\)
- \(c = -2\)

Now, calculating the determinant:
\[ |\text{adj} A| = (-3)(3) - (-1)(-2) = -9 - 2 = -11 \]

### Step 4: Use the Property of Determinants
There is a property of determinants that states:
\[ |\text{adj} A| = |A|^{n-1} \]
where \(n\) is the order of the matrix. For a \(2 \times 2\) matrix, \(n = 2\):
\[ |\text{adj} A| = |A|^{2-1} = |A|^1 = |A| \]
Thus, we have:
\[ |\text{adj} A| = -11 \]

### Final Answer
\[ |\text{adj} A| = -11 \]